Carlos R. P. Hartmann

An optimum symbol-by-symbol decoding rule for linear codes

A decoding rule is presented which minimizes the probability of symbol error over a time-discrete memory]ess channel for any linear error-correcting code when the codewords are equiprobable. The complexity of this rule varies inversely with code rate, making the technique particularly attractive for high rate codes. Examples are given for both block and convolutional codes.

Efficient priority-first search maximum-likelihood soft-decision decoding of linear block codes

The authors present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here converts the decoding problem into a search problem through a graph that is a trellis for an equivalent code of the transmitted code. A generalized Dijkstras algorithm, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to reduce drastically the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. For example, for most real channels of the 35 000 samples tried, simulation results for the (128,64) binary extended BCH code show that the proposed decoding algorithm is fifteen orders of magnitude more efficient in time and in space than that proposed by Wolf (1978). Simulation results for the (104, 52) binary extended quadratic residue code are also given. >

Generalizations of the BCH bound

Cyclic codes generated by polynomials having multiple sets of do — 1 roots in consecutive powers of a nonzero field element are considered and some generalizations of the BCH bound are presented. In particular, it is shown, among other results, that if g(x) GF(q)[x[ is the generator polynomial of a cyclic code Vn of length n such that g(βl+i1c1+i2c2 = 0 for i1 = 0, 1,…, d0 — 2 and i2 = 0,1,…,s, where β GF(qm) is a nonzero element of order n and c1 , c2 are relatively prime to n, then the minimum distance of Vn is at least d0 + s.

An efficient class of unidirectional error detecting/correcting codes

A method for constructing a class of t-error correcting and all unidirectional error-detecting systematic codes is proposed. These codes have been shown to be more efficient than codes constructed using other methods proposed in the literature. In a special case, the code constructed is the Berger code. >

A novel concurrent error detection scheme for FFT networks

A novel algorithm-based fault tolerance scheme is proposed for fast Fourier transform (FFT) networks. It is shown that the proposed scheme achieves 100% fault coverage theoretically. An accurate measure of the fault coverage for FFT networks is provided by taking the roundoff error into account. It is shown that the proposed scheme maintains the low hardware overhead and high throughput of J.Y. Jou and J.A. Abrahams scheme and, at the same time, increases the fault coverage significantly.<<ETX>>

A new approach to the general minimum distance decoding problem: The zero-neighbors algorithm

Minimum distance decoding (MDD) for a general error-correcting linear code is a hard computational problem that recently has been shown to be NP -hard. The complexity of known decoding algorithms is determined by \min (2^{k},2^{n-k}) , where n is the code length and k is the number of information digits. Two new algorithms are suggested that reduce substantially the complexity of MDD. The algorithms use a new concept of zero neighbors--a special set of codewords. Only these codewords (which can be computed in advance) should be stored and used in the decoding procedure. The number of zero neighbors is shown to be very small compared with \min (2^{k},2^{n-k}) for n \gg 1 and a wide range of code rates R = k/n . For example, for R \approx 0.5 this number grows approximately as a square root of the number of codewords.

Efficient Maximum-Likelihood Soft-Decision Decoding of Linear Block Codes Using Algorithm A*

In this paper we present a novel maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here is to convert the decoding problem into a search problem through a graph which is a trellis for an equivalent code of the transmitted code. Algorithm A* is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to drastically reduce the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. Simulation results for the (104, 52) binary extended quadratic residue code and the (128,64) binary extended BCH code are given.

Network Reliability Evaluation Using Probability Expressions

The terminal-pair reliability of a graph (network) is evaluated by means of probability expressions. These expressions result from a transformation of Boolean expressions from the Boolean domain into the probabilistic domain. Basic operations on probability expressions are shown, and a data representation is given for automation of the procedure. The performance of this procedure is compared with other published results. Two new measures, PRI and POST, give relationships between a networks terminal-pair reliability and individual element reliabilities, and derive an element ordering for network diagnosis. These measures can be computed easily using probability expressions.

Some results on the minimum distance structure of cyclic codes

This paper presents a number of interesting results relating to the determination of actual minimum distance of cyclic codes. Codes with multiple sets of consecutive roots are constructed. A bound on the minimum weight of odd-weight codewords is determined. Relations on the distribution of roots of the generator polynomial are investigated. Location polynomials of reversible codes are examined. These results are used to obtain better estimates of the minimum distance of many new cyclic codes.

Generalization of chase algorithms for soft decision decoding of binary linear codes

Soft decision decoding of binary linear block codes transmitted over the additive white Gaussian channel (AWGN) using antipodal signaling is considered. A set of decoding algorithms called generalized Chase algorithms is proposed. In contrast to Chase algorithms, which require a \lfloor (d- 1)/2 \rfloor binary error-correcting decoder for decoding a binary linear block code of minimum distance d , the generalized Chase algorithms can use a binary decoder that can correct less than \lfloor ( d - 1)/2 \rfloor hard errors. The Chase algorithms are particular cases of the generalized Chase algorithms. The performance of all proposed algorithms is asymptotically optimum for high signal-to-noise ratio (SNR). Simulation results for the (47, 23) quadratic residue code indicate that even for low SNR the performance level of a maximum likelihood decoder can be approached by a relatively simple decoding procedure.